Alternating direction multiplier method and low-complexity high-order nonlinear cumulant combined multi-frame reverberation suppression method

Abstract

According to the multi-frame reverberation suppression method based on combination of the alternating direction multiplier method and the low-complexity high-order nonlinear cumulant, underwater reverberation in inter-frame low-rank slow-change components in reverberation is suppressed through the alternating direction multiplier method; water surface reverberation, volume reverberation and noise in inter-frame sparse fast-change components are suppressed by using low-complexity high-order nonlinear cumulants, and continuous moving target bright spots are enhanced, so that an angle-distance sonar two-dimensional image with lower reverberation intensity and clear continuous target bright spots is obtained. Compared with the reverberation suppression performance of an existing low-complexity high-order nonlinear cumulant and multi-frame accumulation after processing of an existing robust principal component analysis method, the multi-frame reverberation suppression method combining the alternative direction multiplier method and the low-complexity high-order nonlinear cumulant has a better reverberation background suppression effect.

Description

Technical Field

[0001] The present invention belongs to the field of signal processing, and in particular relates to a multi-frame reverberation suppression method combining an alternating direction multiplier method with a low-complexity high-order nonlinear cumulant. Background Art

[0002] Active sonar is an active engine for underwater information acquisition. It detects, locates, identifies and images underwater targets by actively emitting sound waves and receiving target echoes. However, during the detection process, reverberation will seriously affect its detection performance.

[0003] Currently, multi-frame reverberation suppression is usually achieved by using the robust principal component analysis (RPCA) method, which uses the correlation between multiple adjacent pulses. For example:

[0004] 1) In the literature (B.Liu, J.Yin, and G.Zhu, “An active detection method for an underwater intruder using the alternating direction method of multipliers,” JAcoust Soc Am, vol. 146, no. 6, 2019.), Liu Bing et al. proposed the alternating direction method of multipliers (ADMM) to detect small underwater targets.

[0005] 2) In the literature (FX Ge, Y. Chen, and W. Li, “Target detection and tracking via structured convex optimization,” in ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing-Proceedings, 2017.), Ge Fengxiang et al. used the Accelerated Proximal Gradient (APG) method to separate reverberation and target highlights, thereby achieving the effect of reverberation suppression.

[0006] 3) In the paper (J. Yin, B. Liu, G. Zhu, and Z. Xie, “Moving target detection using dynamic mode decomposition,” Sensors (Switzerland), vol. 18, no. 10, 2018), Yin Jingwei et al. introduced a dynamic mode decomposition (DMD) method. By calculating and classifying the eigenmodes of an approximate linear model, they ultimately detected the target in the sparse components.

[0007] 4) In the literature (Liu Jianshe, Yin Jingwei, Zhu Guangping, et al. Research on active detection technology of moving targets under ice [J]. Applied Acoustics, 2019, 38(04): 562-568.), Liu Jianshe et al. proposed an inexact augmented Lagrange multiplier (IALM) method to detect moving targets under ice based on low-rank matrix recovery theory;

[0008] However, generally speaking, adjacent multi-frame reverberations can be divided into low-rank, slowly varying components between frames and sparse, quickly varying components between frames. Due to the influence of factors such as noise and multi-frame channel fluctuations, the correlation of reverberations between consecutive frames will decrease, that is, the proportion of low-rank, slowly varying components between frames in the reverberation decreases, and the proportion of sparse, quickly varying components between frames in the reverberation increases. However, the above-mentioned existing reverberation suppression methods using robust principal component analysis do not pay enough attention to the composition and physical properties of reverberation, which will lead to a serious degradation of reverberation suppression performance under conditions of low signal-to-mixture ratio and low inter-frame correlation.

[0009] Therefore, it is urgent to propose a method to improve the reverberation suppression performance under low signal-to-mixture ratio and low inter-frame correlation conditions. Summary of the Invention

[0010] To overcome the inadequate reverberation suppression performance of the aforementioned robust principal component analysis methods under low signal-to-mixture ratio (SMR) and low inter-frame correlation conditions, this paper proposes a multi-frame reverberation suppression method that combines an alternating direction multiplier method with a low-complexity high-order nonlinear cumulant. This method fully considers the reverberation composition and the characteristic differences between reverberation and target highlights, dividing the multi-frame reverberation background into two components: a low-rank, slowly varying component and a sparse, rapidly varying component. It also classifies underwater reverberation as a low-rank, slowly varying component, while surface reverberation, volume reverberation, noise, and target highlights as a sparse, rapidly varying component. First, the alternating direction multiplier method is used to suppress the inter-frame low-rank slow-varying components in multi-frame reverberation, mainly the underwater reverberation, by utilizing the inter-frame correlation features of reverberation and the inter-frame sparse feature differences of target highlights. Then, the inter-frame random features such as surface reverberation, volume reverberation, and noise and the inter-frame continuous feature differences of moving target highlights are utilized to suppress the inter-frame sparse fast-varying components in multi-frame reverberation, mainly the surface reverberation, volume reverberation, noise, etc., by using low-complexity high-order nonlinear cumulants, while effectively enhancing the highlights of continuously moving targets. Finally, the reverberation suppression performance is significantly improved under the conditions of low signal-to-mixing ratio and low inter-frame correlation.

[0011] The technical solution of the present invention is:

[0012] A multi-frame reverberation suppression method combining an alternating direction multiplier method with a low-complexity high-order nonlinear cumulant is characterized by comprising the following steps:

[0013] Step 1: Perform space-time processing on the sonar echo data through matched filtering and beamforming methods to obtain continuous multi-frame detection output;

[0014] Step 2: Discretize the continuous multi-frame detection output obtained in step 1 to obtain a continuous multi-frame data matrix;

[0015] Step 3: Using the inter-frame correlation features of reverberation and the inter-frame sparse feature differences of target highlights, the alternating direction multiplier method is used to perform low-rank sparse matrix decomposition on the continuous multi-frame data matrix obtained in step 2 to obtain a continuous multi-frame low-rank matrix and a continuous multi-frame sparse matrix;

[0016] Step 4: Discard the low-rank matrix of consecutive multiple frames and retain the sparse matrix of consecutive multiple frames to suppress the underwater reverberation in the low-rank slow-changing components between frames. Then, through inverse quantization, the sparse matrix of consecutive multiple frames is converted into a sparse tensor. The sparse tensor is then processed using low-complexity high-order nonlinear cumulants to suppress the surface reverberation, volume reverberation, and noise in the sparse fast-changing components between frames and enhance the highlights of the continuously moving target.

[0017] Step 5: The matrix processed by the low-complexity high-order nonlinear cumulant is used as the final output of reverberation suppression.

[0018] Furthermore, the specific implementation process of step 1 is as follows:

[0019] Step 1.1: Initialize sonar echo data;

[0020] Step 1.2: Perform matched filtering on the initialized sonar echo data and the transmitted signal;

[0021] Step 1.3: Perform beamforming on the output of the matched filter;

[0022] Step 1.4: Normalize the output of beamforming, i.e., the output of continuous multi-frame detection.

[0023] Further,

[0024] The matched filtering output obtained in step 1.2 is:

[0025]

[0026] Where,

[0027] y MF is the output of the matched filter;

[0028] x(t) is the expression of the array receiving signal (equivalent to the initialized sonar echo data);

[0029] s(t) is the expression of the transmitting signal of active sonar;

[0030] τ1 is the time delay between the transmitted signal and the signal received by the array;

[0031] * is for conjugation;

[0032] x n (k) is the received echo of the nth array element, k = 1, 2, ..., N s -1 indicates the number of k-th sampling points, N s is the total number of sampling points within the transmission pulse time;

[0033] The beamforming output obtained in step 1.3 is:

[0034] P(θ)=|W*y MF |

[0035] Where,

[0036] is the beamforming output before normalization, where N θ Represents the angle dimension of a frame of angle-range sonar two-dimensional image, N r It represents the distance dimension of a frame of angle-range sonar 2D image, and N represents the number of frames;

[0037] is the weighting coefficient of conventional beamforming, where Q is the number of elements in the receiving array, is the steering vector, where f0 is the center frequency of the transmitted signal, q=1,2,...,Q, r q =(x q ,y q ) is the qth element position of the receiving array, u(θ) = (sinθ, cosθ) is the unit vector of the beam direction, where θ represents the azimuth angle of the beam direction, c is the speed of sound in water, and j is an imaginary unit;

[0038] The continuous multi-frame detection output obtained in step 1.4 is:

[0039]

[0040] Where,

[0041] is the normalized conventional beamforming output, i.e., the continuous multi-frame detection output;

[0042] max N P(θ) is the maximum value of P(θ) over multiple consecutive frames.

[0043] Furthermore, the specific implementation process of step 2 is as follows:

[0044] First, each frame data matrix in the continuous multi-frame detection output obtained in step 1 is stretched into a column vector and combined in the order of the frames. The column vector is expressed as:

[0045]

[0046] Then, concatenate the N column vectors into a continuous multi-frame data matrix, namely:

[0047]

[0048] Furthermore, the specific implementation process of step 3 is as follows:

[0049] Step 3.1: Input the matrix M and the regularization parameter λ, initialize the continuous multi-frame sparse matrix E0, the continuous multi-frame low-rank matrix A0 and the Lagrange multiplier Y0;

[0050] Step 3.2: To ensure the effective decomposition of the inter-frame sparse fast-varying components and the inter-frame low-rank slow-varying components in the reverberation, an all-one matrix J is compensated to the continuous multi-frame data matrix M;

[0051] Step 3.3: Alternately update the continuous multi-frame low-rank matrix A, the continuous multi-frame sparse matrix E, and the Lagrange multiplier Y;

[0052] Step 3.4: Determine whether the convergence condition is met. If so, output the continuous multi-frame low-rank matrix and continuous multi-frame sparse matrices And end the ADMM cyclic update. If not satisfied, proceed to step 3.3.

[0053] Furthermore, in step 3.1, the low-rank sparse matrix decomposition problem is transformed into the following optimization problem:

[0054]

[0055] Where,

[0056] ||A|| * is the nuclear norm of the continuous multi-frame low-rank matrix A, that is, where σ l is the lth singular value of matrix A;

[0057] ||E||1 is the L1 norm of the continuous multi-frame sparse matrix E, that is where e i,j is the element in the i-th row and j-th column of the matrix E;

[0058] Regularization parameter λ>0.

[0059] Furthermore, the compensated matrix obtained in step 3.2 is:

[0060] D=M+aJ

[0061] Where,

[0062] is the continuous multi-frame data matrix after compensation;

[0063] a=max(M) / 2 is a constant, where max(M) means taking the element with the largest value in the matrix M;

[0064] represents a matrix of all ones.

[0065] Furthermore, the specific implementation process of step 3.3 is as follows:

[0066] 1) Use the soft threshold operator to process the singular value matrix S. The specific formula is:

[0067]

[0068] Where,

[0069] s i,j Represents the element in the i-th row and j-th column of the singular value matrix S;

[0070] a is a constant;

[0071] 2) Fixed E k , Y k , update A k+1 ;

[0072]

[0073] Where,

[0074] U is the left singular vector matrix; V is the right singular vector matrix; S is the singular value matrix;

[0075] E k is the continuous multi-frame sparse matrix before the k+1th cycle;

[0076] Y k is the Lagrange multiplier before the k+1th cycle;

[0077] A k+1 is the continuous multi-frame low-rank matrix obtained in the k+1th cycle;

[0078] k+1 is the number of ADMM cycles;

[0079] μ k is a constant, and when k = 0, μ0 = 1;

[0080] svd(·) is the singular value decomposition of the matrix;

[0081] 3) Use the updated A k+1 , fixed Y k , update E k+1 ;

[0082]

[0083] Where,

[0084] E k+1 is the continuous multi-frame sparse matrix obtained in the k+1th cycle;

[0085] 4) Use the updated A k+1 and E k+1 , update Y k+1 ;

[0086] Y k+1 =Y k +μ k (DA k+1 -E k+1 )

[0087] Where,

[0088] Y k+1 is the Lagrange multiplier obtained in the k+1th cycle;

[0089] 5) Define the initial residual PR and the dual residual DR, and update the constant μ k+1 :

[0090]

[0091] Where,

[0092] PR k+1 =DA k+1 -E k+1 , is the square of the Frobenius norm of the initial residual in the k+1th cycle;

[0093] DR k+1 =μ k (E k+1 -E k ), is the square of the Frobenius norm of the dual residual in the k+1th cycle;

[0094] τ=1.5,ρ=2,κ=σ max (D) / max(N r N θ ,N), where σ max (D) means taking the maximum singular value of the matrix D, and max(·) means taking the maximum value;

[0095] 6) Let k=k+1.

[0096] Furthermore, in step 3.4:

[0097] The convergence condition is:

[0098]

[0099] μ k (E k+1 -E k )<10 -2

[0100] Where,

[0101] is the square of the Frobenius norm of the continuous multi-frame data matrix after compensation, that is, where d m,n is the element in the mth row and nth column of matrix D;

[0102] The continuous multi-frame low-rank matrix output It mainly contains inter-frame low-rank slowly varying components, mainly underwater reverberation components;

[0103] The continuous multi-frame sparse matrix output It mainly includes inter-frame sparse fast-changing components, mainly water surface reverberation, volume reverberation, noise, and target highlights.

[0104] Furthermore, the final output matrix is:

[0105]

[0106] Where,

[0107] P a,b is a sparse tensor The pixel value at (a, b) in multiple consecutive frames;

[0108] Represents the mean value of the pixel value at (a, b) over multiple consecutive frames;

[0109] o is the order of the low-complexity high-order nonlinear cumulant;

[0110] Process the output of a sparse tensor for low-complexity high-order nonlinear cumulants.

[0111] The beneficial effects of the present invention are:

[0112] The present invention suppresses the underwater reverberation in the low-rank slow-varying components between frames in the reverberation by using the alternating direction multiplier method, and suppresses the surface reverberation, volume reverberation and noise in the sparse fast-varying components between frames by using low-complexity high-order nonlinear cumulants, thereby enhancing the bright spots of continuously moving targets, thereby obtaining an angle-range sonar two-dimensional image with lower reverberation intensity and clear bright spots of continuous targets.

[0113] The basic principles and implementation scheme of the present invention have been verified by computer numerical simulations, and the results show that compared with the reverberation suppression performance of multi-frame accumulation after processing by existing robust principal component analysis methods (ADMM, APG, DMD and IALM) and existing low-complexity high-order nonlinear cumulants, the multi-frame reverberation suppression method proposed in the present invention, which combines the alternative directional multiplier method with low-complexity high-order nonlinear cumulants, has better reverberation background suppression effect. BRIEF DESCRIPTION OF THE DRAWINGS

[0114] The above or additional aspects and advantages of the present invention will become apparent and easily understood from the description of the embodiments in conjunction with the following drawings, in which:

[0115] Figure 1 This is a flowchart of the overall implementation of the method proposed in the present invention;

[0116] Figure 2 This is a flowchart for obtaining continuous multi-frame detection output in step 1 of the present invention;

[0117] Figure 3This is a flowchart for implementing the discretization of continuous multi-frame detection output in step 2 of the present invention;

[0118] Figure 4 This is a flowchart for implementing the low-rank sparse decomposition of the matrix in step 3 of the present invention;

[0119] Figure 5 Flowchart for implementing step 4 of the present invention;

[0120] Figure 6 (a) is the angle-range sonar two-dimensional image after matched filtering and beamforming processing. Figure 6 (b) is a continuous target bright spot Figure 6 Enlarged view in (a);

[0121] Figure 7 (a) is the angle-range sonar two-dimensional image after low-complexity high-order nonlinear cumulant processing. Figure 7 (b) is a continuous target bright spot Figure 7 Enlarged view in (a);

[0122] Figure 8 (a) is the angle-range sonar two-dimensional image after DMD processing and multi-frame accumulation. Figure 8 (b) is a continuous target bright spot Figure 8 Enlarged view in (a); Figure 8 (c) is the angle-range sonar two-dimensional image after multiple frames of accumulation after APG processing. Figure 8 (d) is the continuous target bright spot Figure 8 (c) Enlarged view; Figure 8 (e) is the angle-range sonar two-dimensional image after IALM processing and multi-frame accumulation, Figure 8 (f) is the continuous target bright spot Figure 8 (e) Enlarged image; Figure 8 (g) is the angle-range sonar two-dimensional image after ADMM processing and accumulation of multiple frames. Figure 8 (h) is the continuous target bright spot Figure 8 Enlarged image in (g);

[0123] Figure 9 (a) is the angle-range sonar two-dimensional image processed by the multi-frame reverberation suppression method using the alternating direction multiplier method combined with the low-complexity high-order nonlinear cumulant. Figure 9 (b) is a continuous target bright spot Figure 9 Enlarged view in (a);

[0124] Figure 10The results of calculating the integrated sidelobe ratio deviation between consecutive frames using the multi-frame reverberation suppression methods using low-complexity high-order nonlinear cumulants, multi-frame accumulation after DMD processing, multi-frame accumulation after APG processing, multi-frame accumulation after IALM processing, multi-frame accumulation after ADMM processing, and the alternating direction multiplier method combined with low-complexity high-order nonlinear cumulants.

[0125] Figure 11 This is a distance dimension slice diagram of the multi-frame reverberation suppression method using the alternating direction multiplier method combined with low-complexity high-order nonlinear cumulant when the angle of the continuous target bright spot is 60°;

[0126] Figure 12 Schematic diagram of multi-frame accumulation. DETAILED DESCRIPTION

[0127] The multi-frame reverberation suppression method of the present invention, which combines the alternating direction multiplier method with low-complexity high-order nonlinear cumulants, achieves its reverberation background suppression effect through the following three aspects:

[0128] 1. Considering the differences in reverberation composition, reverberation, and target highlights, the multi-frame reverberation background is divided into two parts: low-rank slow-varying components between frames and sparse fast-varying components between frames. The underwater reverberation is classified as the low-rank slow-varying components between frames, while the surface reverberation, volume reverberation, noise, and target highlights are classified as sparse fast-varying components between frames.

[0129] 2. Based on the reverberation composition, physical characteristics and division method, a multi-frame reverberation suppression process was constructed:

[0130] (1) Using matched filtering and beamforming to perform space-time processing on multi-frame sonar echo data, a continuous multi-frame detection output is obtained, and then the continuous multi-frame detection output is discretized to obtain a continuous multi-frame data matrix;

[0131] (2) Using the alternating direction multiplier method, the continuous multi-frame data matrix is ​​subjected to low-rank sparse matrix decomposition to obtain a continuous multi-frame low-rank matrix and a continuous multi-frame sparse matrix. The continuous multi-frame low-rank matrix mainly contains inter-frame low-rank slow-varying components, mainly underwater reverberation components; the continuous multi-frame sparse matrix mainly contains inter-frame sparse fast-varying components, mainly surface reverberation, volume reverberation, noise, and target bright spots.

[0132] (3) Discard the continuous multi-frame low-rank matrix and retain the continuous multi-frame sparse matrix to suppress the bottom reverberation in the low-rank slow-changing components between frames. Then, through inverse quantization, the continuous multi-frame sparse matrix is ​​converted into a sparse tensor, and the sparse tensor is processed using a low-complexity high-order nonlinear cumulant to suppress the water surface reverberation, volume reverberation, and noise in the sparse fast-changing components between frames, and enhance the highlights of the continuous moving target.

[0133] (4) The matrix processed by low-complexity high-order nonlinear cumulant is used as the final output of reverberation suppression;

[0134] 3. Through computer numerical simulation, the results of multi-frame reverberation suppression methods using multi-frame accumulation after DMD processing, multi-frame accumulation after IALM processing, multi-frame accumulation after APG processing, multi-frame accumulation after ADMM processing, low-complexity high-order nonlinear cumulant processing, and alternating direction multiplier method combined with low-complexity high-order nonlinear cumulant are respectively given. This proves that the method proposed in the present invention achieves better reverberation background suppression performance than the existing robust principal component analysis methods (DMD, IALM, APG, ADMM) after multi-frame accumulation and the existing low-complexity high-order nonlinear cumulant processing under the conditions of low signal-to-mixing ratio and low inter-frame correlation.

[0135] The present invention will be further described below with reference to the accompanying drawings.

[0136] like Figure 1 As shown, the multi-frame reverberation suppression method proposed in the present invention combines the alternating direction multiplier method with the low-complexity high-order nonlinear cumulant. The specific implementation process is as follows:

[0137] Step 1: If Figure 2 As shown in the figure, the sonar echo data is processed in space and time by the matched filtering and beamforming method to obtain continuous multi-frame detection output. The specific implementation process is as follows:

[0138] Step 1.1: Initialize sonar echo data. The sonar echo data used in the present invention is collected from field experiments or generated by simulation experiments.

[0139] Step 1.2: Perform matched filtering on the initialized sonar echo data and the transmitted signal. Specifically:

[0140] For a multi-base sonar, let the expression of the active sonar transmission signal be s(t), and the expression of the array reception signal (equivalent to the initialized sonar echo data) be x(t). Perform matched filtering on the active sonar transmission signal and the array reception signal:

[0141]

[0142] Where,

[0143] y MF is the output of the matched filter;

[0144] t is the time variable;

[0145] τ1 is the time delay between the transmitted signal and the signal received by the array;

[0146] * is for conjugation;

[0147] x n (k) is the received echo of the nth array element, k = 1, 2, ..., N s -1 indicates the number of k-th sampling points, N s is the total number of sampling points within the transmission pulse time;

[0148] Step 1.3: Perform beamforming on the output of the matched filter. Taking conventional beamforming as an example, specifically:

[0149] Assume the steering vector is a(θ), and its expression is:

[0150]

[0151] Where,

[0152] f0 is the center frequency of the transmitted signal;

[0153] r q =(x q ,y q ) is the qth element position of the receiving array, where q = 1, 2, ..., Q, and Q is the total number of elements in the receiving array;

[0154] u(θ)=(sinθ,cosθ) is the unit vector of the beam direction, where θ represents the azimuth angle of the beam direction;

[0155] c is the speed of sound in water;

[0156] j is the imaginary unit;

[0157] Then the weighting coefficients of conventional beamforming are:

[0158]

[0159] Where,

[0160] W is the weighting coefficient of conventional beamforming;

[0161] The output of conventional beamforming is:

[0162] P(θ)=|W*y MF |

[0163] Where,

[0164] is the beamforming output before normalization, where N θ Represents the angle dimension of a frame of angle-range sonar two-dimensional image, N r It represents the distance dimension of a frame of angle-range sonar 2D image, and N represents the number of frames;

[0165] Step 1.4: Normalize the beamforming output, i.e., the continuous multi-frame detection output. Specifically:

[0166] Normalize P(θ):

[0167]

[0168] Where,

[0169] is the normalized conventional beamforming output, i.e., the continuous multi-frame detection output;

[0170] max N P(θ) is the maximum value of P(θ) over multiple consecutive frames.

[0171] Step 2: If Figure 3 As shown, the continuous multi-frame detection output obtained in step 1 is discretized to obtain a continuous multi-frame data matrix. Specifically:

[0172] First, each frame data matrix in the continuous multi-frame detection output obtained in step 1 is stretched into a column vector and combined in the order of the frames. The column vector is expressed as:

[0173]

[0174] Then, concatenate the N column vectors into a continuous multi-frame data matrix, namely:

[0175]

[0176] Step 3: If Figure 4 As shown in the figure, the inter-frame correlation features of reverberation and the inter-frame sparse feature differences of target highlights are used to perform low-rank sparse matrix decomposition on the continuous multi-frame data matrix using the alternating direction method of multipliers (ADMM) to obtain the continuous multi-frame low-rank matrix and the continuous multi-frame sparse matrix. The specific implementation process is as follows:

[0177] Step 3.1: Input the matrix M and the regularization parameter λ, initialize the continuous multi-frame sparse matrix E0, the continuous multi-frame low-rank matrix A0 and the Lagrange multiplier Y0, and transform the low-rank sparse matrix decomposition problem into the following optimization problem:

[0178]

[0179] Where,

[0180] ||A|| * is the nuclear norm of the continuous multi-frame low-rank matrix A, that is, where σ lis the lth singular value of matrix A;

[0181] ||E||1 is the L1 norm of the continuous multi-frame sparse matrix E, that is where e i,j is the element in the i-th row and j-th column of the matrix E;

[0182] Regularization parameter λ>0;

[0183] Step 3.2: To ensure the effective decomposition of the inter-frame sparse fast-varying components and the inter-frame low-rank slow-varying components in the reverberation, an all-one matrix J is compensated to the continuous multi-frame data matrix M. The compensated matrix is:

[0184] D=M+aJ

[0185] Where,

[0186] is the continuous multi-frame data matrix after compensation;

[0187] a=max(M) / 2 is a constant, where max(M) means taking the element with the largest value in the matrix M;

[0188] represents a matrix of all 1s;

[0189] Step 3.3: Alternately update the continuous multi-frame low-rank matrix A, the continuous multi-frame sparse matrix E, and the Lagrange multiplier Y;

[0190] 1) Use the soft threshold operator to process the singular value matrix S. The specific formula is:

[0191]

[0192] Where,

[0193] s i,j Represents the element in the i-th row and j-th column of the singular value matrix S;

[0194] a is a constant;

[0195] 2) Fixed E k , Y k , update A k+1 ;

[0196]

[0197] Where,

[0198] U is the left singular vector matrix; V is the right singular vector matrix; S is the singular value matrix;

[0199] E k is the continuous multi-frame sparse matrix before the k+1th cycle;

[0200] Y k is the Lagrange multiplier before the k+1th cycle;

[0201] A k+1 is the continuous multi-frame low-rank matrix obtained in the k+1th cycle;

[0202] k+1 is the total number of ADMM cycles;

[0203] μ k is a constant, and when k = 0, μ0 = 1;

[0204] svd(·) is the singular value decomposition of the matrix;

[0205] 3) Use the updated A k+1 , fixed Y k , update E k+1 ;

[0206]

[0207] Where,

[0208] E k+1 is the continuous multi-frame sparse matrix obtained in the k+1th cycle;

[0209] 4) Use the updated A k+1 and E k+1 , update Y k+1 ;

[0210] Y k+1 =Y k +μ k (DA k+1 -E k+1 )

[0211] Where,

[0212] Y k+1 is the Lagrange multiplier obtained in the k+1th cycle;

[0213] 5) Define the initial residual PR and the dual residual DR, and update the constant μ k+1 :

[0214]

[0215] Where,

[0216] PR k+1 =DA k+1 -E k+1 , is the square of the Frobenius norm of the initial residual in the k+1th cycle;

[0217] DR k+1 =μ k (E k+1 -E k ), is the square of the Frobenius norm of the dual residual in the k+1th cycle;

[0218] τ=1.5,ρ=2,κ=σ max (D) / max(N r N θ ,N), where σ max (D) means taking the maximum singular value of the matrix D, and max(·) means taking the maximum value;

[0219] 6) Let k = k + 1;

[0220] Step 3.4: Determine whether the convergence condition is met. If so, output the continuous multi-frame low-rank matrix and continuous multi-frame sparse matrices And end the ADMM cycle update. If it is not satisfied, continue to step 3.3;

[0221] The convergence conditions are as follows:

[0222]

[0223] μ k (E k+1 -E k )<10 -2

[0224] Where,

[0225] is the square of the Frobenius norm of the continuous multi-frame data matrix after compensation, that is, where d m,n is the element in the mth row and nth column of matrix D;

[0226] The continuous multi-frame low-rank matrix obtained by the above process mainly contains low-rank slow-varying components between frames, mainly underwater reverberation components; while the continuous multi-frame sparse matrix mainly contains sparse fast-varying components between frames, mainly surface reverberation, volume reverberation, noise, and target highlights.

[0227] Step 4: If Figure 5 As shown in the figure, the continuous multi-frame low-rank matrix A is discarded and the continuous multi-frame sparse matrix E is retained to suppress the bottom reverberation in the low-rank slowly varying components between frames; then the continuous multi-frame sparse matrix is ​​converted into Convert to sparse tensor Use low-complexity high-order nonlinear cumulants to process sparse tensors to suppress water surface reverberation, volume reverberation, and noise in sparse fast-changing components between frames, and enhance the highlights of continuously moving targets;

[0228] The calculation formula for the low-complexity high-order nonlinear cumulant processing sparse tensor ε is:

[0229]

[0230] Where,

[0231] P a,b is the pixel value at (a, b) in the sparse tensor ε over multiple consecutive frames;

[0232] Represents the mean value of the pixel value at (a, b) over multiple consecutive frames;

[0233] o is the order of the low-complexity high-order nonlinear cumulant;

[0234] Process the output of a sparse tensor for low-complexity high-order nonlinear cumulants.

[0235] Step 5: The matrix processed by low-complexity high-order nonlinear cumulant As the final output of reverberation suppression.

[0236] The proposed method can be used to robustly detect frogmen, unmanned underwater vehicles (UUVs), surface vessels, submarines, marine life, and other moving targets in highly reverberant environments. To simplify the description, this implementation example uses a typical underwater, slow-moving, small target detection process as an example to verify the effectiveness of the proposed method. This example utilizes computer simulation to perform numerical simulations to verify the effectiveness of the proposed method.

[0237] The basic simulation parameters are as follows: the transmitting array element is a single hydrophone, the receiving array is a 64-element circular array, the waveform of the transmitted signal is a linear frequency modulation signal (LFM), the center frequency of the LFM signal is 100 kHz, the bandwidth is 15 kHz, the duration of a detection cycle is 0.2 s, the pulse width of a single transmitted signal is 0.01 s, the sampling frequency is 57 kHz, and the array element spacing is 7.5 × 10 -3 m, the scanning angle is 0°~120°, the scanning interval is 1°, the signal-to-mixing ratio is set to -20dB, and the inter-frame correlation is 0.6.

[0238] Target parameters: The target's initial angle is 60°, the initial distance is 110m, and it moves in a uniform straight line at a speed of 1m / frame.

[0239] Image parameters: Output angle-range sonar 2D image, sampling angle dimension is 121, sampling distance dimension is 11400, and frame number is 10. Select the continuous target bright spots accumulated from frame 1 to frame 10 as the output result.

[0240] For the three robust principal component analysis methods ADMM, APG and IALM, the regularization coefficients are all set to The initial value of the penalty coefficient is 1, and the maximum number of iterations is K. max =60.

[0241] When using low-complexity high-order nonlinear cumulants to process sparse tensors, the selected order o=3.9.

[0242] according to Figure 2 The sonar echo data in the simulation example is processed by the following process, such as Figure 6 As shown in Figure 1, in the angle-range sonar two-dimensional image after matched filtering and beamforming processing, the continuous target bright spots are "swamped" by the reverberation.

[0243] Sonar echo data can be processed in three ways:

[0244] (1) Only the existing robust principal component analysis methods (DMD, APG, IALM and ADMM) are used to process the sonar echo data, and the multi-frame output results are accumulated, such as Figure 12 As shown, multiple frames of images are accumulated into one frame. The above processing methods are respectively referred to as "multi-frame accumulation after DMD processing", "multi-frame accumulation after APG processing", "multi-frame accumulation after IALM processing", and "multi-frame accumulation after ADMM processing". The results are shown in Figure 8 As shown in the figure, in the multi-frame accumulated images after DMD processing, the target continuous bright spots are still "swamped" by strong reverberation; in the multi-frame accumulated images after IALM processing and APG processing, the target continuous bright spots can be vaguely observed by the naked eye, and there are a lot of strong reverberations that are not suppressed around the bright spots, and the reverberation suppression performance is roughly the same; in the multi-frame accumulated images after ADMM processing, the target continuous bright spots can be clearly observed by the naked eye, but there are some reverberations with different intensities and relatively random distribution around the bright spots;

[0245] (2) Only low-complexity high-order nonlinear cumulants are used to process the sonar echo data. The above processing method is referred to as "low-complexity high-order nonlinear cumulants processing", such as Figure 7 As shown in the figure, the continuous target bright spots are clearly visible in the image processed by low-complexity high-order nonlinear cumulant, but there is still strong reverberation around them.

[0246] (3) According to Figure 2 、 3, 4 and 5, a multi-frame reverberation suppression method combining the alternating direction multiplier method with the low-complexity high-order nonlinear cumulant is used to process the sonar echo data, and the multi-frame reverberation suppression method combining the alternating direction multiplier method with the low-complexity high-order nonlinear cumulant is referred to as the “proposed method”. Figure 9 As shown in , the continuous target bright spots in the image processed by the proposed method are clearly visible, and there is almost no reverberation observable to the naked eye in the area near the target bright spots. Figure 11 As shown in FIG, compared with the traditional method using matched filtering and beamforming, the method proposed in the present invention improves the strong reverberation background by at least 20 dB.

[0247] Figure 10 The results of the integrated sidelobe ratio deviation under continuous multi-frame conditions are shown in Figure 2, respectively. The integrated sidelobe ratio deviation values ​​obtained by the proposed method are significantly higher than those obtained by the other five processing methods at the same frame number. Specifically, the values ​​are at least 7dB higher than those obtained by the multi-frame accumulation after ADMM processing, at least 3dB higher than those obtained by the nonlinear cumulant processing, at least 17dB higher than those obtained by the multi-frame accumulation after IALM processing and the multi-frame accumulation after APG processing, and at least 14dB higher than those obtained by the multi-frame accumulation after DMD processing. This demonstrates the effectiveness and superiority of the proposed method in suppressing reverberation.

[0248] The above description is only a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any technician familiar with this technical field can easily think of various equivalent modifications or replacements within the technical scope disclosed in the present invention, and these modifications or replacements should all be included in the scope of protection of the present invention.

Claims

1. A multi-frame reverberation suppression method combining an alternating direction multiplier method with a low-complexity high-order nonlinear cumulant, characterized in that: The specific steps include: Step 1: Perform space-time processing on the sonar echo data through matched filtering and beamforming methods to obtain continuous multi-frame detection output; Step 2: Discretize the continuous multi-frame detection output obtained in step 1 to obtain a continuous multi-frame data matrix; Step 3: Using the inter-frame correlation features of reverberation and the inter-frame sparse feature differences of target highlights, the alternating direction multiplier method is used to perform low-rank sparse matrix decomposition on the continuous multi-frame data matrix obtained in step 2 to obtain a continuous multi-frame low-rank matrix and a continuous multi-frame sparse matrix; Step 4: Discard the low-rank matrix of consecutive multiple frames and retain the sparse matrix of consecutive multiple frames to suppress the underwater reverberation in the low-rank slow-changing components between frames. Then, through inverse quantization, the sparse matrix of consecutive multiple frames is converted into a sparse tensor. The sparse tensor is then processed using low-complexity high-order nonlinear cumulants to suppress the surface reverberation, volume reverberation, and noise in the sparse fast-changing components between frames and enhance the highlights of the continuously moving target. Step 5: The matrix processed by the low-complexity high-order nonlinear cumulant is used as the final output of reverberation suppression.

2. The multi-frame reverberation suppression method combining an alternating direction multiplier method with a low-complexity high-order nonlinear cumulant as claimed in claim 1, characterized in that: The specific implementation process of step 1 is as follows: Step 1.1: Initialize sonar echo data; Step 1.2: Perform matched filtering on the initialized sonar echo data and the transmitted signal; Step 1.3: Perform beamforming on the output of the matched filter; Step 1.4: Normalize the output of beamforming, i.e., the output of continuous multi-frame detection.

3. The multi-frame reverberation suppression method combining an alternating direction multiplier method with a low-complexity high-order nonlinear cumulant as claimed in claim 2, characterized in that: The matched filtering output obtained in step 1.2 is: Where, y MF is the output of the matched filter; x(t) is the expression of the array receiving signal (equivalent to the initialized sonar echo data); s(t) is the expression of the transmitting signal of active sonar; τ1 is the time delay between the transmitted signal and the signal received by the array; * is for conjugation; x n (k) is the received echo of the nth array element, k = 1, 2, ..., N s -1 indicates the number of k-th sampling points, N s is the total number of sampling points within the transmission pulse time; The beamforming output obtained in step 1.3 is: P(θ)=|W*y MF | Where, is the beamforming output before normalization, where N θ Represents the angle dimension of a frame of angle-range sonar two-dimensional image, N r It represents the distance dimension of a frame of angle-range sonar 2D image, and N represents the number of frames; is the weighting coefficient of conventional beamforming, where Q is the number of elements in the receiving array, is the steering vector, where f0 is the center frequency of the transmitted signal, q=1,2,...,Q, r q =(x q ,y q ) is the qth element position of the receiving array, u(θ) = (sinθ, cosθ) is the unit vector of the beam direction, where θ represents the azimuth angle of the beam direction, c is the speed of sound in water, and j is an imaginary unit; The continuous multi-frame detection output obtained in step 1.4 is: Where, is the normalized conventional beamforming output, i.e., the continuous multi-frame detection output; max N P(θ) is the maximum value of P(θ) over multiple consecutive frames.

4. The multi-frame reverberation suppression method combining an alternating direction multiplier method with a low-complexity high-order nonlinear cumulant as claimed in claim 1, characterized in that: The specific implementation process of step 2 is as follows: First, each frame data matrix in the continuous multi-frame detection output obtained in step 1 is stretched into a column vector and combined in the order of the frames. The column vector is expressed as: Then, concatenate the N column vectors into a continuous multi-frame data matrix, namely:

5. The multi-frame reverberation suppression method combining an alternating direction multiplier method with a low-complexity high-order nonlinear cumulant as claimed in claim 1, characterized in that: The specific implementation process of step 3 is as follows: Step 3.1: Input the matrix M and the regularization parameter λ, initialize the continuous multi-frame sparse matrix E0, the continuous multi-frame low-rank matrix A0 and the Lagrange multiplier Y0; Step 3.2: To ensure the effective decomposition of the inter-frame sparse fast-varying components and the inter-frame low-rank slow-varying components in the reverberation, an all-one matrix J is compensated to the continuous multi-frame data matrix M; Step 3.3: Alternately update the continuous multi-frame low-rank matrix A, the continuous multi-frame sparse matrix E, and the Lagrange multiplier Y; Step 3.4: Determine whether the convergence condition is met. If so, output the continuous multi-frame low-rank matrix and continuous multi-frame sparse matrices And end the ADMM cyclic update. If not satisfied, proceed to step 3.

3.

6. The multi-frame reverberation suppression method combining an alternating direction multiplier method with a low-complexity high-order nonlinear cumulant as claimed in claim 5, characterized in that: In step 3.1, the low-rank sparse matrix decomposition problem is transformed into the following optimization problem: Where, ||A|| * is the nuclear norm of the continuous multi-frame low-rank matrix A, that is, where σ l is the lth singular value of matrix A; ||E||1 is the L1 norm of the continuous multi-frame sparse matrix E, that is where e i,j is the element in the i-th row and j-th column of the matrix E; Regularization parameter λ>0.

7. The multi-frame reverberation suppression method combining an alternating direction multiplier method with a low-complexity high-order nonlinear cumulant as claimed in claim 5, characterized in that: The compensated matrix obtained in step 3.2 is: D=M+aJ Where, is the continuous multi-frame data matrix after compensation; a=max(M) / 2 is a constant, where max(M) means taking the element with the largest value in the matrix M; represents a matrix of all ones.

8. The multi-frame reverberation suppression method combining an alternating direction multiplier method with a low-complexity high-order nonlinear cumulant as claimed in claim 5, characterized in that: The specific implementation process of step 3.3 is as follows: 1) Use the soft threshold operator to process the singular value matrix S. The specific formula is: Where, s i,j Represents the element in the i-th row and j-th column of the singular value matrix S; a is a constant; 2) Fixed E k , Y k , update A k+1 ; Where, U is the left singular vector matrix; V is the right singular vector matrix; S is the singular value matrix; E k is the continuous multi-frame sparse matrix before the k+1th cycle; Y k is the Lagrange multiplier before the k+1th cycle; A k+1 is the continuous multi-frame low-rank matrix obtained in the k+1th cycle; k+1 is the number of ADMM cycles; μ k is a constant, and when k = 0, μ0 = 1; svd(·) is the singular value decomposition of the matrix; 3) Use the updated A k+1 , fixed Y k , update E k+1 ; Where, E k+1 is the continuous multi-frame sparse matrix obtained in the k+1th cycle; 4) Use the updated A k+1 and E k+1 , update Y k+1 ; Y k+1 =Y k +μ k (DA k+1 -E k+1 ) Where, Y k+1 is the Lagrange multiplier obtained in the k+1th cycle; 5) Define the initial residual PR and the dual residual DR, and update the constant μ k+1 : Where, PR k+1 =DA k+1 -E k+1 , is the square of the Frobenius norm of the initial residual in the k+1th cycle; DR k+1 =μ k (E k+1 -E k ), is the square of the Frobenius norm of the dual residual in the k+1th cycle; τ=1.5,ρ=2,κ=σ max (D) / max(N r N θ ,N), where σ max (D) means taking the maximum singular value of the matrix D, and max(·) means taking the maximum value; 6) Let k=k+1.

9. The multi-frame reverberation suppression method combining an alternating direction multiplier method with a low-complexity high-order nonlinear cumulant as claimed in claim 5, characterized in that: In step 3.4: The convergence condition is: μ k (AND k+1 -AND k )<10 -2 Where, is the square of the Frobenius norm of the continuous multi-frame data matrix after compensation, that is, where d m,n is the element in the mth row and nth column of matrix D; The continuous multi-frame low-rank matrix output It mainly contains inter-frame low-rank slowly varying components, mainly underwater reverberation components; The continuous multi-frame sparse matrix output It mainly includes inter-frame sparse fast-changing components, mainly water surface reverberation, volume reverberation, noise, and target highlights.

10. The multi-frame reverberation suppression method combining an alternating direction multiplier method with a low-complexity high-order nonlinear cumulant as claimed in claim 1, characterized in that: The final output matrix is: Where, P a,b is a sparse tensor The pixel value at (a, b) in multiple consecutive frames; Represents the mean value of the pixel value at (a, b) over multiple consecutive frames; o is the order of the low-complexity high-order nonlinear cumulant; Process the output of a sparse tensor for low-complexity high-order nonlinear cumulants.

Images